Weeks 1,2,3,4,5,6,7,8

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Cartesian Product Context: characterized by finding all possible pairings between all possible pairings between 2 or more sets of objects.

Investigating Quantity

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This is the last topic of the course! Hooray!!

Addition of integers

Subtraction of integers

Multiplication of integers

Decimals

Percents

Properties of Modular Arithmatic(Multiplication)

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Commutative Property of multiplication:axb(mod 5)=bxa(mod 5)

Closure Property of Multiplication

Commutative Property of Multiplication

Identity Property of Multiplication

Inverse Property of Multiplication

Properties of Modular Arithmatic(Addition)

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Commutative Property of Addition:a+b(mod 5)=b+a(mod 5)

Closure Property of Addition

Communative Property of Addition

Identity Property of Addition

Proportional Reasoning

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Relative thinking: identifying relationships between two quantities and comparing them.

Utilizing

Ratio Sense

Rational Numbers

Recognition of quantities and how they change

Relative thinking.

Proportion

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a/b = c/d

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Where two ratios are equal.

Ratio

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Example of ratio: 2:3

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Comparing two quantities regardless of whether units are the same.

Division of fractions

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When using the partition model it's most important to know the # of groups as well as the size.

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Repeated Subtraction

Partition

Multiplication of fractions

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An area model is best used as a table.

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Area Model

Fractions

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Cuisennaire rods were an interesting manipulative to use to comprehend fractions.

Interpretations of a fraction

Part-Whole interpretation

Copies

Division

Separating groups

Ratios

Comparing 2 separate things

Rational Numbers

Any number that can be expressed as the quotient of two integers. a/b

Use of Cuisennaire Rods

Problem Solving

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Don't forget to read over the problem over a second time so that you understand what you're reading.

Understand the Problem

Devise A Plan

Implement you Plan

Sequences

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Geometric Sequences are sequences of numbers with a common ratio; that is, if you form the ratio of any two consecutive terms in the sequence the ratio is the same.

Arithmetic Sequence

Geometric Sequence

Recurrence Relationship Sequence

Sets

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Equivalent Sets and Equal sets are not always the same.

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Venn Diagram

Universal Set

Conjoined Set

Complements

Equivalent Sets

Equal Sets

Number Systems

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The Mayan system is probably the most difficult number system to use.

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Tally System

Egyptian System

Myan System

Babylonian System

Roman System

Hindu-Arabic System

Models

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The Discrete Model is characterized by combining sets of two discrete objects.

Discrete Model

Counted Quantities

Continuous Model

Measured Quantities

Investigating Quantities

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Cubes!

Bases

Cubes

a^3

Flats

a^2

Longs

a^1

Units

1Unit

Addition

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Identity Property of Division was probably the most difficult to understand.

Closure Property of Addition

If aeX and beX then a+beX

Communative Property of Whole Numbers

If aeW and beW then a+b=b+a

Associative Property of Whole Numbers

If aeW, beW and ceW, then (a+b)+c=a+(b+c)=(a+c)+b

Identity Property of Division of Whole Numbers

If aeW then a+o=a=o+a

Division Models/Contexts

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Divison Models/Contexts

Partition: Equal Sharing

Characterizing by distributing a given quantity among a specified number of groups(partition) and determining the size(amount) in each group(partition).

Know: Quantity of #'s starting with, and # of groups

Find: Size of each group

Measurement

Characterized by using a given quantity to create groups(partitions) of a specified size(amount) and determining the # of groups that are formed.

Know: Quantity, and size of each group

Find: # of groups

Multiplication Models/Contexts

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Distributive Property: aeW, beW, ceW, then a(b+c) = ab+ac

Associative Property (grouping)

(axb)xc = ax(bxc)

Communicative Property

If aeW and beW, then ab = ba

Closure Property

If aeW and beW, then abeW

Identity Property

aeW, 1xa=a, 1 is the identity element

Zero Property

aeW, oxa=o

Distributive Property

aeW, beW, ceW, then a(b+c) = ab+ac

Subtraction Models/Contexts

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Subtraction is characterized by starting with some initial quantity and removing a specified amount.

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Closure Property (doesn't always work)

If aeW and beW, then (a-b)eW

Communative Property (doesn't always work)

If aeW and beW then a-b = b-a

Identity Property

If aeW then a-o = o-a

Number Theory

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Prime numbers are only divisible by one and themselves.

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Odd Numbers

An even number + 1

Even Numbers

Multiples of 2

Prime Numbers

Only divisible by 1 and itself

Composite Numbers

Has a positive factor other than 1 and itself

Prime Factorization

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Prime factorization is a factorization containing only prime numbers.

Fundamental Theorem of Arithmetic

Means you can't find more than one prime factorization for a number.